2 BELL SYSTEM TECHNICAL JOURNAL 



filter. Formulas are deduced and a set of representative curves 

 computed and plotted which show the dependence of the building-up 

 process on the constants and number of sections of the filter and the 

 frequency of the applied e.m.f. The outstanding deduction from 

 this phase of the problem is that as the selectivity of the filter is 

 increased either by narrowing the transmission band or increasing the 

 number of sections, the time required for sinusoidal currents to build 

 up is proportionally increased. This fact, it may be remarked, sets a 

 theoretical limit to the amount of selectivity which can be employed 

 in communication circuits. 



3. The character and duration of the transient current ivhen a sinusoidal 

 voltage outside the frequency tra?ismission range is applied to the filter. 

 It will be found that in this case a transient disturbance penetrates 

 the filter which is enormous compared to the final steady state. The 

 magnitude of this disturbance decreases very slowly with the number 

 of filter sections and its duration increases therewith. This phenome- 

 non is an important special case of the general limitations of the 

 selectivity of the filter in the transient state. 



4. The energy which penetrates through selective circuits from random 

 interference. The energy spectrum of random interference, that is, 

 interference from random disturbances is discussed and a formula is 

 deduced which defines the figure of merit of a selective circuit with 

 respect to random interference. This formula leads to general deduc- 

 tions of practical importance regarding the relative merits of selective 

 networks in the transient state and their inherent limitations. It 

 also provides a method for experimentally determining the spectrum 

 of random interference. 



Unfortunately the complexity of transient phenomena is such as 

 to absolutely require a large amount of mathematical analysis. Con- 

 sequently, while the mathematics has been relegated as far as possi- 

 ble to Appendices, a considerable amount necessarily appears in the 

 text. The writers, however, have endeavored to emphasize the 

 physical significance of the mathematics and have included only 

 that which is absolutely essential to an understanding of the problem 

 and the appropriate methods of attack. 



In order to keep the analysis within manageable limits and in a 

 form to admit of relatively simple and instructive interpretation, 

 the formulas will be restricted for the most part to non-dissipative 

 filters and the effects of terminal reflections will be ignored. 2 These 



2 The general solution for the case of arbitrary terminal impedances is given in 

 Appendix IV and briefly discussed. 



